On the stability of a class of non-linear continuous systems†

A procedure to find the existence and domain of asymptotic stability for a class of systems described by a set of non-linear partial differential equations is presented in the context of semigroup theory.

No a priori bound is imposed on the non-linearity which is assumed to be a quad ratic (bilinear) or multilinear form.

The linearized system can be non-self-adjoint and need not be asymptotically stable equation by equation.

The solution of the stability problem is carried out. in two steps :

(a) the development of an L₂-like norm that bounds above the sup norm, a requirement to obtain a useful estimate of the non-linear term, and

(a)the computation of an estimate of the L₂-likc norm of the semigroup generated by the linearized system in terms of its usual L₂ norm.     

On the stability of control systems with randomly time-varying characteristics†

The purpose of this paper is to study the stability conditions of randomly time-varying control systems.

First we define the stability, in the stoehastic sense, of the control system with randomly time-varying characteristics and secondly we obtain the conditions for global stability in the mean for a control system which is governed in each time interval by two different differential equations with probability p and 1-q respectively.

Finally we examine the stability condition for the concrete control systems by experiments of the Monte Carlo method.     

On the stability problem for distributed parameter energy systems†

In this paper asymptotic stability of the non-trivial or operating equilibrium state of a distributed parameter energy system is considered. Sufficient conditions are obtained by the method of comparison functions. This enables one to obtain bounds on the maximum value of the state variables. The criteria derived depend on the eigenvalues of the linearized system as well as on a bound for the feedback control law.     

Stability of a class of interconnected systems†

Sufficient conditions are given for exponential stability of the equilibrium of systems of interconnected components taken from a class of nonlinear, time-varying, multi-terminal, differential, algebraic and mixed components. The analysis is based on the mathematical models of the unconstrained components and the linear constraints imposed by the interconnections between the components. A Liapunov function is constructed for the interconnected system from Liapunov functions for the individual components. A parameter vector appearing in the Liapunov function is selected in a prescribed optimal manner. The Liapunov function so constructed can be used to establish a lower bound on the rate of decay of the system. An example of a ninth-order, non-linear system is included to demonstrate the application to design.     

On the stability of time-varying systems using an observer for feedback control†

The effects upon stability are considered when an observer is incorporated in a time-varying system to estimate the immeasurable state variables in order to implement the feedback control law.     

On the bounds of the trajectories of differential systems†

The behaviour of differential systems is investigated by considering the stability and instability of such systems with respect to certain sub-sets of the state space These Sets may in general be time-varying, and their properties do not only yield information about the stability of a system but also estimates of the bounds of the system trajectories. In all cases the results which are established yield sufficient conditions for stability and instability, and in general involve the existence of Lyapunov-like functions which do not appear to possess the usual definiteness requirements on V and [Vdot].

The developed theory is applied to two special cases: in one case, only time-invariant sub-sets are considered: in the second case, the time interval [t₀, ∞) over which the systems me defined is truncated to [t ₀, t _a  + T), T < ∞, So the that case of stability over a finite time interval may be considered.     

On the controllability of distributed parameter systems†

The reachable states of distributed parameter systems where the magnitude of the control energy is constrained are found. The systems considered are described by linear partial differential equations on a bounded domain. The controls may be either distributed or at the boundary and are required to satisfy a constraint in the magnitude of their norm. The results are then used to find conditions for null controllability.     

On the asymptotic stability of a particular class of distributed parameter feedback control systems†

In this paper the stability problem for a particular class of non-linear feedback control systems governed by a parabolic partial differential equation is considered. A sufficient condition for global asymptotic stability of the null equilibrium state is derived by the method of comparison functions.     

On practical stability†

In this paper, a class of non-linear non-autonomous systems with multiple non-linearities is considered. Sufficient conditions are developed for a type of practical stability with specified settling time. The conditions are independent of the actual Form of the non-linear characteristics so that they can be interpreted as conditions for ‘absolute’ practical stability. The stability test is reduced to verification of the Hurwitz property of a constant matrix. This makes the stability analysis of the considered class of non-linear systems convenient for machine computations

The proposed practical stability analysis is applied to a third-order system with several non-linearities.     

Connective stability of large-scale stochastic systems†

In this work, we will introduce the notion of connective stability in the mean for large-scale stochastic systems under structural perturbations. Sufficient conditions for this kind of stability are obtained by the decomposition-aggregation method in the context of vector Liapuiiov functions.     

On the design of optimal output feedback control systems†

For a linear control system with quadratic performance index the optimal control takes a feedback form of all state variables. However, if there are some states which are not fed in the control system, it is impossible to obtain the optimal feedback control by using the usual mathematical optimization technique such as dynamic programming or the maximum principle.

This paper presents the optimal control of output feedback systems for a quadratic performance index by using a new parameter optimization technique.

Since the optimal feedback gains depend on the initial states in the output feedback control system, two cases where (1) the initial states are known, and (2) the statistical properties of initial states such as mean and covariance matrices are known, are considered here. Furthermore, the proposed method for optimal output feedback control is also applied to sampled-data systems.     

On optimization of non-linear stochastic systems†

Results of application of the perturbation approach along with stochastic approximations to the problem of optimizing non-linear stochastic systems subject to quadratic index are discussed. The perturbation approach is first introduced with the help of linear regulator systems. This is combined subsequently with statistical linearization in order to derive sub-optimal solutions for the problem of optimizing a non-linear system. Finally, a new stochastic approximation technique, that permits more accuracy than the linearization, is suggested for the non-linear problem. Numerical examples are presented to compare the results of the various approximation methods and also to illustrate the possible advantages of non-linearization.     

On linear modelling of non-linear systems†

In many cases of interest, systems described by non-linear differential equations are modelled by linear equations for purposes of synthesizing feedback controllers. A theorem relating the stability properties of the linear model and the non-linear system is presented and discussed.     

On the design of optimal and sub-optimal control systems†

Though dynamic programming is an excellent and versatile algorithm, there are many cases in which we cannot apply this algorithm favourably to control system design because of the limited capacity of a digital computer and other difficulties.

Considering these facts, the successively optimizing method and the combined algorithm of dynamic programming and successively optimizing method are presented in this paper.

One of the objects of this paper is to present a practical method for obtaining the optimal solution of control systems with an integral form performance index by the use of dynamic programming.

The other is to present sub-optiinal solutions by the successively optimizing method and the combined algorithm.

These methods are very easy when compared with dynamic programming. Furthermore, these sub-optimal solutions can be made to converge to the optimum by using the recurrent equations repeatedly.     

On the controllability of systems with a time-varying delay†

This paper is concerned with the controllability of a system with a time-varying delay. The system to be considered here is described by a linear differential-difference equation of a retarded type, where a time-varying delay is a certain class of a time-varying function. First, the concepts of controllability for linear differential-difference systems with a constant delay, introduced by Weiss (1987), are developed to the system with a time-varying delay. Second, necessary and sufficient conditions for controllability are obtained. Finally, the controllability of a stationary system with a constant delay is treated, including the results of Chyung and Lee (1966).     

Non-asymptotic stability of complex valued differential systems†

Sufficient conditions for the existence of orbital motion and non-asymptotic, non-orbital motion in the solutions of constant coefficient complex variable differential equations are obtained through Liapunov's direct method. Matrix analogues of those conditions assure the corresponding type of motion of interconnected systems whose complex coefficient matrices are simultaneously diagonalizable.     

On the boundedness of the solutions of some systems with retarded arguments†

The boundedness of the solutions of delay-differential systems with large retardation is proved. The result is obtained by comparing with a system whose exact solutions are obtained.     

Switched control of frequency of interconnected power systems using variable structure†

This paper proposes a new technique based on eigenvalue assignment for the design of a variable structure load frequency controller for interconnected power systems. An important feature of the variable structure controller is that it can effectively improve the transient performance of the system while at the same time ensuring zero steady-state error. Further, when the variable structure system is operated in the so-called sliding mode, the response of the system becomes insensitive to plant parameter variations. The control problem is formulated and the results of a digital simulation study are presented for illustration. For comparison purposes, results obtained using a conventional integral controller and a linear optimal controller are also included.     

On delay-dependent robust stability of neutral systems

The delay-dependent robust stability of uncertain linear neutral systems with delays is investigated. Both discrete-delay-dependent/neutral-delay-independent and neutral-/discrete- delay-dependent stability criteria will be developed. The proposed stability criteria are formulated in the form of linear matrix inequalities and it is easy to check the robust stability of the considered systems. By introducing certain Lyapunov-Krasovskii functional the mathematical development of our result avoids model transformation and bounding for cross terms, which lead to conservatism. Finally, numerical example is given to indicate the improvement over some existing results.     

Stability,transient behaviour and trajectory bounds of interconnected systems†

In practice, many systems are complex and of high dimension. Many such systems may be viewed as being composed of several simpler sub-systems which when connected in an appropriate fashion yield the original composite system. The stability, the transient behaviour and estimates for the trajectory bounds of certain composite systems are analysed in terms of their sub-systems. This is accomplished by defining the stability of sub-systems and of composite systems in terms of certain time-varying sub-sets of the state space which are pre-specified in a given problem.

After stating definitions of stability for sub-systems which are under the influence of perturbing forces and for composite systems, theorems are stated and proved which yield sufficient conditions for stability. These theorems involve the existence of Lyapunov-like functions which do not possess any particular definiteness requirements on V and [Vdot].

The time-varying sub-sets of the state space which are utilized in the stability definitions and which arise in conjunction with the stability theorems yield estimates of the transient behaviour and of the trajectory bounds of both sub-systems and composite systems.

To demonstrate the generality of the developed theory, several special cases are considered. Also, some specific examples are worked out to demonstrate the methods involved.